Intuitively, if we are guaranteed that \(|\operatorname{tr}(P \rho )| > \tilde{\epsilon }\)  and \(N_2\)  is sufficiently large, then the majority vote will concentrate on the correct answer. Hence, the quantum state \(\rho ^{\otimes N_2}\)  is approximately contained in one of the two eigenspaces of the observable \(E\) . As a result, after measuring \(E\) , the quantum state \(\rho ^{\otimes N_2}\)  would remain approximately the same.
This allows us to keep measuring the sign of \(\operatorname{tr}(P \rho )\)  for many different Pauli operators.
This strategy is a key element in the original protocol for shadow tomography [1]}.
The rigorous guarantee is given by Lemma REF , which relies on the quantum union bound [2]}.
